A couple that come to mind for me:
Fastlove by Adele at the Grammy’s. She (and Hans Zimmer) took an up tempo George Michael song and turned it into a haunting song shortly after George Michael’s death. I thought of it again after Adele’s recent special and can’t get it out of my mind!
Hurt Johnny Cash
So, so many
Whitney Houston - ‘I Will Always Love You’
Jeff Buckley - ‘Hallelujah’
Adele - ‘Make You Feel My Love’
Chris Cornell (with honorable mention to Sinead O’Connor) – ‘Nothing Compares 2 U’
Cyndi Lauper - ‘Girls Just Want to Have Fun’
Manfred Mann’s Earth Band - ‘Blinded By the Light’
Joan Jett and The Blackhearts - ‘I Love Rock ‘n’ Roll’
Jimi Hendrix - ‘All Along the Watchtower’
Aretha Franklin - ‘Respect’
Tina Turner - ‘Proud Mary’
Janis Joplin - ‘Me and Bobby McGee’
Agree with these two 
@skieurope, but Cohen’s voice just does it for me on the original. Houston is definitely better than Parton even though she wrote it.
And Aretha…does ANYone do anything better?
Even Dolly admits that Whitney’s is better. But it does not bother her since she got even more wealthy from the royalties. 
And for the record, I also like Dolly’s version. And many OG versions - Springsteen comes to mind - are also iconic in their own right. So no disrespect to the original artists.
Kui Lee, “Days of Your Youth”
Black Magic Woman- Santana. The percussion adds the perfect vibe to the song missing in the original.
Freedy Johnston “Witchita Lineman”
Doobie Brothers - Jesus is Just Alright
Pearl Jam - Baba O’Riley
Pearl Jam - Fortunate Son
The Bangles - Hazy Shade of Winter
Molly Hatchet - Dreams I’ll Never See
Reckless Kelly-Castanets
Frank Turner- Last Christmas
Kathleen Edwards- Money Talks
Last Train Home- I wish it would rain
Kirsty MacColl- A New England (bonus fact- she got Billy Bragg to write an extra verse for her and since her death in 2000, he always sings the verse)
Grace Petrie- Fairytale of New York (she fixes the problematic line)
Sally Timms-- Cry, Cry, Cry
Cowboy Junkies-- Sweet Jane
Jason Isbell- Driver 8
Cry Cry Cry-- Fall on Me
Great Big Sea-- Its The End of The World as We Know It
“Dancing Days” by Stone Temple Pilots (Led Zeppelin song)
“Time of the Season” by Dave Matthews Band (Zombies song)
Lil Nas X Jolene
and agree Johnny Cash’s version of Hurt is the best
And while they aren’t covers I know and love most of Weird Al’s songs from the Food Album better than the originals.
Jolene - Sara Gazarek
Love the one you’re with - Luther Vandross
Somewhere over the rainbow (Live) - Trisha Yearwood
The Sound of Silence - Disturbed
Hallelujah - KD Lang
Ave Maria - Chris Botti
Amazing Grace - Pentatonix
Jingle Bell Rock - Juice Newton
Christmas Canon - Trans-Siberian Orchestra (cover of Canon in D major)
Fields of Gold by Eva Cassidy
One of the loviest songs I’ve ever heard. Tragic that Eva died so young.
Change is Gonna Come by Justin Rolon of the PS22 Choir
In Times Like These Mavis Staples
Twist and Shout - The Beatles
Brandi Carlile - “Take Me Home, Country Roads”
Chris Cornell - “Nothing Compares 2 U”
Lana Del Rey - “Doin’ Time”
Linkin Park - “Rolling in the Deep”
Metallica - “Turn the Page” and “Whiskey in the Jar”
I love a good cover song!
The Chicks: Landslide
John Mayer: Free Fallin
Captain Fantastic Cast: Sweet Child of Mine
Nirvana: Man Who Sold the World
Pearl Jam: Last Kiss
Johnny Cash: Hurt
Adele: Lovesong
Jasmine Thompson: Mad World
Basically 90% of From the Screen to Your Stereo (1 & 2, not 3) from Newfound Glory.
There were also a few songs on Glee that I legitimately liked better than the originals.
I also really like Johnny Cash - Hurt. And …
Hallelujah - Brandi Carlile
A Whiter Shade of Pale - Annie Lennox
Zombie - Bad Wolves
The Sound of Silence - Disturbed
Both good choices.
IMO, the original is still the best, but my favorite cover of this song is by The White Stripes.
Agree about Eva Cassidy. Beautiful voice.
I also like Israel Kamakawiwoʻole’s renditions of Somewhere Over the Rainbow and What a Wonderful World (even though the originals are perfect).